A set $S\subseteq \re^n$ is called to be {\it Semidefinite (SDP)} representable if $S$ equals the projection of a set in higher dimensional space which is describable by some Linear Matrix Inequality (LMI). The contributions of this paper are: (i) For bounded SDP representable sets $W_1,...,W_m$, we give an explicit construction of an SDP representation for $\cv{\cup_{k=1}^mW_k}$. This provides a technique for building global SDP representations from the local ones. (ii) For the SDP representability of a compact convex semialgebraic set $S$, we prove sufficient condition: the boundary $\bdS$ is positively curved, and necessary condition: $\bdS$ has nonnegative curvature at smooth points and on nondegenerate corners. This amounts to the strict versus nonstrict quasi-concavity of defining polynomials on those points on $\bdS$ where they vanish. The gaps between them are $\bdS$ having positive versus nonnegative curvature and smooth versus nonsmooth points. A sufficient condition bypassing the gaps is when some defining polynomials of $S$ are sos-concave. (iii) For the SDP representability of the convex hull of a compact nonconvex semialgebraic set $T$, we find that the critical object is $\pt_cT$, the maximum subset of $\pt T$ contained in $\pt \cv{T}$. We prove sufficient conditions for SDP representability: $\pt_cT$ is positively curved, and necessary conditions: $\pt_cT$ has nonnegative curvature at smooth points and on nondegenerate corners. The gaps between them are similar to case (ii). The positive definite Lagrange Hessian (PDLH) condition is also discussed.
Consider the closed convex hull $K$ of a monomial curve given parametrically as $(t^{m_1},\ldots,t^{m_n})$, with the parameter $t$ varying in an interval $I$. We show, using constructive arguments, that $K$ admits a lifted semidefinite description by $\mathcal{O}(d)$ linear matrix inequalities (LMIs), each of size $\left\lfloor \frac{n}{2} \right\rfloor +1$, where $d= \max \{m_1,\ldots,m_n\}$ is the degree of the curve. On the dual side, we show that if a univariate polynomial $p(t)$ of degree $d$ with at most $2k+1$ monomials is non-negative on $\mathbb{R}_+$, then $p$ admits a representation…
Given a subset $\mathbf{S}=\{A_1, \dots, A_m\}$ of $\mathbb{S}^n$, the set of $n \times n$ real symmetric matrices, we define its {\it spectrahull} as the set $SH(\mathbf{S}) = \{p(X) \equiv (Tr(A_1 X), \dots, Tr(A_m X))^T : X \in \mathbfΔ_n\}$, where ${\bf Δ}_n$ is the {\it spectraplex}, $\{ X \in \mathbb{S}^n : Tr(X)=1, X \succeq 0 \}$. We let {\it spectrahull membership} (SHM) to be the problem of testing if a given $b \in \mathbb{R}^m$ lies in $SH(\mathbf{S})$. On the one hand when $A_i$'s are diagonal matrices, SHM reduces to the {\it convex hull membership} (CHM), a fundamental problem i…
In scheduling and timetabling applications, the mutual-exclusion constraint stipulates that certain pairs of tasks that cannot be executed at the same time. This corresponds to the vertex colouring problem in graph theory, for which there are well-known semidefinite programming (SDP) relaxations. In practice, however, the mutual-exclusion constraint is typically combined with many other constraints, whose SDP representability has not been studied. We present SDP relaxations for a variety of mutual-exclusion scheduling and timetabling problems, starting from a bound on the number of tasks execu…
Quadratically constrained quadratic programs (QCQPs) are a fundamental class of optimization problems well-known to be NP-hard in general. In this paper we study sufficient conditions for a convex hull result that immediately implies that the standard semidefinite program (SDP) relaxation of a QCQP is tight. We begin by outlining a general framework for proving such sufficient conditions. Then using this framework, we show that the convex hull result holds whenever the quadratic eigenvalue multiplicity, a parameter capturing the amount of symmetry present in a given problem, is large enough. O…
Quadratically constrained quadratic programs (QCQPs) are a fundamental class of optimization problems. In a QCQP, we are asked to minimize a (possibly nonconvex) quadratic function subject to a number of (possibly nonconvex) quadratic constraints. Such problems arise naturally in many areas of operations research, computer science, and engineering. Although QCQPs are NP-hard to solve in general, they admit a natural convex relaxation via the standard (Shor) semidefinite program (SDP) relaxation. In this tutorial, we will study the SDP relaxation for general QCQPs, present various exactness con…
Northwestern University（西北大学）；University of California Berkeley（加州大学伯克利分校）；University of Southern California（南加州大学）
We consider the convex quadratic optimization problem with indicator variables and arbitrary constraints on the indicators. We show that a convex hull description of the associated mixed-integer set in an extended space with a quadratic number of additional variables consists of a single positive semidefinite constraint (explicitly stated) and linear constraints. In particular, convexification of this class of problems reduces to describing a polyhedral set in an extended formulation. While the vertex representation of this polyhedral set is exponential and an explicit linear inequality descri…
Consider a finite system of non-strict polynomial inequalities with solution set $S\subseteq\mathbb R^n$. Its Lasserre relaxation of degree $d$ is a certain natural linear matrix inequality in the original variables and one additional variable for each nonlinear monomial of degree at most $d$. It defines a spectrahedron that projects down to a convex semialgebraic set containing $S$. In the best case, the projection equals the convex hull of $S$. We show that this is very often the case for sufficiently high $d$ if $S$ is compact and "bulges outwards" on the boundary of its convex hull. Now le…
Quadratically constrained quadratic programs (QCQPs) are a fundamental class of optimization problems well-known to be NP-hard in general. In this paper we study conditions under which the standard semidefinite program (SDP) relaxation of a QCQP is tight. We begin by outlining a general framework for proving such sufficient conditions. Then using this framework, we show that the SDP relaxation is tight whenever the quadratic eigenvalue multiplicity, a parameter capturing the amount of symmetry present in a given problem, is large enough. We present similar sufficient conditions under which the…
We show {\it semidefinite programming} (SDP) feasibility problem is equivalent to solving a {\it convex hull relaxation} (CHR) for a finite system of quadratic equations. On the one hand, this offers a simple description of SDP. On the other hand, this equivalence makes it possible to describe a version of the {\it Triangle Algorithm} for SDP feasibility based on solving CHR. Specifically, the Triangle Algorithm either computes an approximation to the least-norm feasible solution of SDP, or using its {\it distance duality}, provides a separation when no solution within a prescribed norm exists…
Quadratically constrained quadratic programs (QCQPs) are a highly expressive class of nonconvex optimization problems. While QCQPs are NP-hard in general, they admit a natural convex relaxation via the standard (Shor) semidefinite program (SDP) relaxation. Towards understanding when this relaxation is exact, we study general QCQPs and their (projected) SDP relaxations. We present sufficient (and in some cases, also necessary) conditions for objective value exactness (the condition that the objective values of the QCQP and its SDP relaxation coincide) and convex hull exactness (the condition th…
A closed convex conic subset $\mathcal{S}$ of the positive semidefinite (PSD) cone is rank-one generated (ROG) if all of its extreme rays are generated by rank-one matrices. The ROG property of $\mathcal{S}$ is closely related to the exactness of SDP relaxations of nonconvex quadratically constrained quadratic programs (QCQPs) related to $\mathcal{S}$. We consider the case where $\mathcal{S}$ is obtained as the intersection of the PSD cone with finitely many homogeneous linear matrix inequalities and conic constraints and identify sufficient conditions that guarantee that $\mathcal{S}$ is ROG.…
